Multivariate shrinkage estimation of small area means and proportions

The familiar (univariate) shrinkage estimator of a small area mean or propo rtion combines information from the small area and a national survey. We de fine a multivariate shrinkage estimator which combines information also acr oss subpopulations and outcome variables. The superiority of the multivaria te shrinkage over univariate shrinkage, and of the univariate shrinkage ove r the unbiased (sample) means, is illustrated on examples of estimating the local area rates of economic activity in the subpopulations defined by eth nicity, age and sex. The examples use the sample of anonymized records of i ndividuals from the 1991 UK census. The method requires no distributional a ssumptions but relies on the appropriateness of the quadratic loss function . The implementation of the method involves minimum outlay of computing. Mu ltivariate shrinkage is particularly effective when the area level means ar e highly correlated and the sample means of one or a few components have sm all sampling and between-area variances. Estimation for subpopulations base d on small samples can be greatly improved by incorporating information fro m subpopulations with larger sample sizes.